is equal to the ſquare of DE. and therefore the accelerations generated in the paſſage of the bodies from D and I to F and K are equal. Therefore the velocities of the bodies in E and K are alſo equal: and by the ſame reaſoning they will always be found equal in any ſubſequent equal diſtances. Q. E. D.
By the ſame reaſoning, bodies of equal velocities and equal diſtances from the centre will be equally retarded in their aſcent to equal diſtances. Q. E. D.
Cor. 1. Therefore if a body either oſcillates by hanging to a ſtring, or by any poliſhed and perfectly ſmooth impediment is forced to move in a curve line; and another body aſcends or deſcends in a right line, and their velocities be equal at any one equal altitude; their velocities will be alſo equal at all other equal altitudes. For, by the ſtring of the pendulous body, or by the impediment of a veſſel perfectly ſmooth, the ſame thing will be effected, as by the tranſverſe force NT. The body is neither accelerated nor retarded by it, but only is obliged to quit its rectilinear courſe.
Cor. 2. Suppoſe the quantity P to be the greateſt diſtance from the centre to which a body can aſcend, whether it be oſcillating, or revolving in a trajectory, and ſo the ſame projected upwards from any point of a trajectory with the velocity it has in that point. Let the quantity A be the diſtance of the body from the centre in any other point of the orbit; and let the centripetal force be always as the power of the quantity A, the index of which power n-1, is any number n diminiſhed by unity. Then the velocity in every altitude A will be as , and therefore will be given. For by prop. 59. the velocity of
